It is well known that vari­ous stat­ist­ics of a large sample (of size \( n \)) are ap­prox­im­ately dis­trib­uted ac­cord­ing to the nor­mal law. The asymp­tot­ic ex­pan­sion of the dis­tri­bu­tion of the stat­ist­ic in a series of powers of \( n^{-1/2} \) with a re­mainder term gives the ac­cur­acy of the ap­prox­im­a­tion. H. Cramér [1937] first ob­tained the asymp­tot­ic ex­pan­sion of the mean, and re­cently P. L. Hsu [1945] has ob­tained that of the vari­ance of a sample. In the present pa­per we ex­tend the Cramér–Hsu meth­od to Stu­dent’s stat­ist­ic. The the­or­em proved states es­sen­tially that if the pop­u­la­tion dis­tri­bu­tion is non-sin­gu­lar and if the ex­ist­ence of a suf­fi­cient num­ber of mo­ments is as­sumed, then an asymp­tot­ic ex­pan­sion can be ob­tained with the ap­pro­pri­ate re­mainder. The first four terms of the ex­pan­sion are ex­hib­ited in for­mula (35).

Re­new­al the­ory has been treated by many pure and ap­plied math­em­aticians. Among the former we may men­tion Feller, Täck­lind and Doob. The prin­cip­al lim­it the­or­em (for one-di­men­sion­al, pos­it­ive, lat­tice ran­dom vari­ables) was however proved earli­er by Kolmogorov in 1936 as the er­god­ic the­or­em for de­nu­mer­able Markov chains. A par­tial res­ult for the non-lat­tice case was first proved by Doob us­ing the the­ory of Markov pro­cesses, and the com­plete res­ult by Black­well. The ex­ten­sion of the re­new­al the­or­em to ran­dom vari­ables tak­ing both pos­it­ive and neg­at­ive val­ues was first giv­en by Wolfow­itz and the au­thor [Chung and Wolfow­itz 1952], for the lat­tice case. A par­tial res­ult for the non-lat­tice case, us­ing a purely ana­lyt­ic­al ap­proach, was ob­tained by Pol­lard and the au­thor [Chung and Pol­lard 1952]. For the lit­er­at­ure see [Chung and Wolfow­itz 1952].

Asymp­tot­ic prop­er­ties are es­tab­lished for the Rob­bins–Monro [1951] pro­ced­ure of stochastic­ally solv­ing the equa­tion \( M(x) = \alpha \). Two dis­joint cases are treated in de­tail. The first may be called the “bounded” case, in which the as­sump­tions we make are sim­il­ar to those in the second case of Rob­bins and Monro. The second may be called the “quasi-lin­ear” case which re­stricts \( M(x) \) to lie between two straight lines with fi­nite and non­van­ish­ing slopes but pos­tu­lates only the bounded­ness of the mo­ments of \( Y(x) - M(x) \). In both cases it is shown how to choose the se­quence \( \{a_n\} \) in or­der to es­tab­lish the cor­rect or­der of mag­nitude of the mo­ments of \( x_n - \theta \). Asymp­tot­ic nor­mal­ity of \( a^{1/2}_n(x_n - \theta) \) is proved in both cases un­der a fur­ther as­sump­tion. The case of a lin­ear \( M(x) \) is dis­cussed to point up oth­er pos­sib­il­it­ies. The stat­ist­ic­al sig­ni­fic­ance of our res­ults is sketched.

Let \( \{X_i\},\ i=1,2,\dots \) be a se­quence of in­de­pend­ent and identic­ally dis­trib­uted in­teg­ral val­ued ran­dom vari­ables such that 1 is the ab­so­lute value of the greatest com­mon di­visor of all val­ues of \( x \) for which \( P(X_i=x) > 0 \). Define \( S_n=\sum_{i=1}^n X_i \). Chung and Fuchs [1951] showed that if \( x \) is any in­teger, \( S_n=x \) in­fin­itely of­ten with prob­ab­il­ity 1 ac­cord­ing as \( EX_i=0 \) or \( \neq 0 \), provided that \( E|X_i| < \infty \). Let \( 0 < EX_i < \infty \), and \( A \) de­note a set of in­tegers con­tain­ing an in­fin­ite num­ber of pos­it­ive in­tegers. It will be shown that any such set \( A \) will be vis­ited in­fin­itely of­ten with prob­ab­il­ity 1 by the se­quence \( \{S_n\}\ n=1,2,\dots \). Con­di­tions are giv­en so that sim­il­ar res­ults hold for the case where \( X_i \) has a con­tinu­ous dis­tri­bu­tion and the set \( A \) is a Le­besgue meas­ur­able set whose in­ter­sec­tion with the pos­it­ivfe real num­bers has in­fin­ite Le­besgue meas­ure.

We give a very short proof of the re­cur­rence the­or­em of Chung and Fuchs [1951] in one and two di­men­sions. This new ele­ment­ary proof does not de­tract from the old one which uses a sys­tem­at­ic meth­od based on the char­ac­ter­ist­ic func­tion and yields a sat­is­fact­ory gen­er­al cri­terion. But the present meth­od, be­sides its brev­ity, also throws light on the com­bin­at­or­i­al struc­ture of the prob­lem.

The pur­pose of this pa­per is to present a simple uni­fied ap­proach to a group of the­or­ems in semi-group the­ory called the “ex­po­nen­tial for­mu­las”, due to Hille, Phil­lips, Wid­der and D. G. Kend­all [Hille and Phil­lips 1957, p. 354]. A more gen­er­al and ap­par­ently new for­mula is ar­rivefd at, which in­cludes some known cases. It turns out that these for­mu­las are in es­sence sum­mab­il­ity meth­ods which are best com­pre­hen­ded from the point of view of ele­ment­ary prob­ab­il­ity the­ory. They are all in the spir­it of S. Bern­stein’s proof of Wei­er­strass’s ap­prox­im­a­tion the­or­em, the same idea be­ing present in M. Riesz’s proof of Hille’s first ex­po­nen­tial for­mula (see [Hille et al. 1957, p. 314]). Where­as the de­tails here are just a little sim­pler than in [Hille and Phil­lips 1957], it seems of some in­terest to ex­hib­it the gen­er­al pat­tern and to re­duce the proofs to routine veri­fic­a­tions. The read­er who is not ac­quain­ted with the lan­guage of prob­ab­il­ity should have no dif­fi­culty in everything in­to the lan­guage of clas­sic­al ana­lys­is. But the prob­ab­il­ity way of think­ing is really ger­mane to the sub­ject.

Doob’s ver­sion of the fun­da­ment­al con­ver­gence the­or­em of po­ten­tial the­ory as­serts that if \( (f_n) \) is a de­creas­ing se­quence of ex­cess­ive func­tion and \( f \) is the su­per­me­di­an func­tion \( \inf_n f_n \), then the set where \( f \) dif­fers from \( \hat{f} \) (its reg­u­lar­ized func­tion) is semi-po­lar. Many beau­ti­ful proofs of this res­ult are avail­able in the lit­er­at­ure. Here is a trivi­al one.

The Pois­son pro­cess was one of Rényi’s fa­vor­ite top­ics. Here its fa­mil­i­ar prop­er­ties are dis­cussed from the gen­er­al stand­point of re­new­al the­ory, lead­ing to cer­tain simple char­ac­ter­iz­a­tions. Some of the ob­ser­va­tions made be­low are ap­par­ently new des­pite the enorm­ous lit­er­at­ure on the Pois­son pro­cess, oth­ers are stated for pedgo­gic reas­ons–a con­sid­er­a­tion that had al­ways con­cerned Rényi too.

The the­or­ems by Kh­intchine, Koro­ly­uk, and Dobrush­in in the the­ory of sta­tion­ary point pro­cesses are ba­sic and simple the­or­ems. Koro­ly­uk’s the­or­em was ori­gin­ally de­rived from the Palm–Kh­intchine for­mu­las; a dir­ect proof was giv­en in Cramér–Lead­bet­ter [1967]. Its real sim­pli­city seems to be ob­scured by the slightly com­plic­ated present­a­tion of the proof. The same may be said of the proof of Dobrush­in’s the­or­em in­volving an un­ne­ces­sary con­tra­pos­i­tion as well as some ep­si­lon­ics. Both res­ults be­come quiet trans­par­ent when dealt with by stand­ard meth­ods of meas­ure and in­teg­ra­tion in sample space. After all, these are prob­lems of prob­ab­il­ity the­ory and nowadays stu­dents spend a lot of time learn­ing this kind of “ab­stract” set-up. It would be a pity not to use the know­ledge so ac­quired in straight­for­ward situ­ations such as these the­or­ems. In do­ing so we ar­rive at cer­tain nat­ur­al ex­ten­sions which seem to put the res­ults in prop­er per­spect­ive. The res­ults in \( R^d \), ob­tained by the same meth­od, seem to be new.

The joint dis­tri­bu­tion of the time since last exit, and the time un­til next en­trance, in­to a unique bound­ary point is giv­en in the fol­low­ing for­mula:
\[ P\{\gamma(t)\in ds:\ \beta(t)\in du\} = E(ds)\,\theta(u-s)\,du \]
for \( s < t < u \). The bound­ary point may be re­placed by a re­gen­er­at­ive phe­nomen­on.

Let \( \{W(t):t\geq 0\} \) be the stand­ard Browni­an mo­tion with all paths con­tinu­ous. Let \( M(t)= \max_{0\leq s\leq t}W(s) \) be the max­im­um pro­cess and \( Y(t)=M(t)-W(t) \) be re­flect­ing Browni­an mo­tion. If \( d_{\varepsilon}(t) \) is the num­ber of times \( Y \) crosses down from \( \varepsilon \) to 0 be­fore time \( t \), then it was Paul Lévy’s idea that
\begin{equation*}\tag{1}
P\bigl\{\lim_{\varepsilon\to 0} \varepsilon d_{\varepsilon}(t) = M(t),\
\forall t\geq 0\bigr\} = 1.
\end{equation*}
In [1974] Itô and McK­ean demon­strated the al­most sure con­ver­gence of \( \varepsilon d_{\varepsilon}(t) \) us­ing mar­tin­gale meth­ods. To identi­fy the lim­it they used the hard fact, due to
Lévy, that
\begin{equation*}\tag{2}
P\Bigl\{\lim_{\varepsilon\to 0}
\frac{ \text{measure} \{s: Y(s) < \varepsilon,\ s\leq t\} }{2\varepsilon}=M(t),
\ \forall t\geq 0\Bigr\} = 1
\end{equation*}
and com­puted the second mo­ment of the dif­fer­ence of the ex­pres­sions in (1) and (2). In this pa­per, by ex­amin­ing the ex­cur­sions in Browni­an mo­tion and us­ing a new for­mula for the dis­tri­bu­tion of their max­ima, we ob­tain a dir­ect
iden­ti­fic­a­tion of the lim­it in (1) without us­ing (2).

The pur­pose of this note is to es­tab­lish a suf­fi­cient con­di­tion for re­cur­rence of a ran­dom walk \( (S_n) \) in \( R^2 \). It fol­lows from it that if \( S_n/n^{1/2} \) is asymp­tot­ic­ally nor­mal then we have re­cur­rence.

We con­sider, on the group of in­tegers, a ran­dom walk start­ing from the ori­gin and whose steps ad­mit as pos­sible val­ues ex­actly two in­tegers, \( a \) and \( b \), with \( a < 0 < b \). In the par­tic­u­lar case \( a=-1 \), we give an ex­pli­cit ex­pres­sion for the law of the first re­turn time to the ori­gin.

About 1923, the great math­em­atician Paul Lévy in­ven­ted a fam­ily of prob­ab­il­ity dis­tri­bu­tions (laws) called “stable”. If \( X \) and \( Y \) are in­de­pend­ent ran­don vari­ables with the law \( L \), then for any con­stants \( a > 0 \) and \( b \), there ex­ist con­stants \( c > 0 \) and \( d \) such that the law of \( aX + bY \) is the same as \( cZ + d \), where \( Z \) is a ran­don vari­able with the same law \( L \).

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